
// Make newform 125.2.e.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_125_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_125_e_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_125_2_e_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_125_2_e_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [25, -25, -10, 5, 11, -7, 4, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-7800, 1065, 4949, 2103, -1896, 747, -714, 406], [-10265, 1055, 5129, 2297, -1924, 698, -776, 420], [-15715, 2190, 8547, 3584, -3353, 1246, -1327, 728], [18200, -2485, -9470, -4907, 4424, -1743, 1666, -857], [-18960, 3235, 9520, 4991, -4492, 1624, -1623, 891], [-20550, 2415, 9646, 5723, -4891, 1942, -1829, 955]];
    Rf_basisdens := [1, 1, 1355, 1355, 1355, 1355, 1355, 1355];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_125_e();"
function MakeCharacter_125_e()
    N := 125;
    order := 10;
    char_gens := [2];
    v := [1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_125_e_Hecke();"
function MakeCharacter_125_e_Hecke(Kf)
    N := 125;
    order := 10;
    char_gens := [2];
    char_values := [[0, 0, 1, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[1, -1, 0, 0, 0, 0, 0, 0], [1, 0, -1, 1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -2, 0, -1, 0, -1, 0], [0, 0, 0, 0, -2, 0, 0, 0], [1, -1, 0, 0, -1, -1, 0, 0], [-1, 1, 4, -2, 3, 1, 0, 1], [-2, 0, 0, -3, 2, 1, 1, -1], [0, 1, -1, 2, -1, 1, 1, 0], [-3, 4, 5, 0, -1, 2, 2, 1], [0, 1, 0, 4, 0, -2, -3, 2], [-2, -1, 0, 4, -1, -1, -1, 1], [-1, -3, 1, -3, 4, 1, -1, -1], [-3, -1, 1, -5, 6, 0, 1, -1], [0, 0, -1, 0, 3, 3, 0, -2], [2, 0, -2, -1, -6, -3, 0, -2], [3, -1, -4, 1, -3, 3, 1, 1], [-2, -3, -1, 0, -2, -4, 4, -6], [0, -4, -4, 2, -4, -4, 0, -4], [-4, 6, 1, -6, 5, 3, 2, 2], [2, -3, -5, 4, -2, -2, -2, 0], [7, -6, -4, 3, 1, -3, -8, 1], [0, 5, 8, -4, 8, 4, 1, 4], [-6, -1, 5, 0, 7, 0, 0, -2], [12, 0, -4, 9, -7, -4, 0, 4], [10, -3, -12, 12, -2, -4, -1, -1], [1, 0, 2, -3, -1, 3, 0, 5], [-6, 3, 9, -3, 9, 0, 0, 3], [-9, 5, 3, -7, -2, 1, 3, 3], [0, -1, -7, 1, 6, -1, 0, 0], [-4, -2, -2, -4, 2, -2, -2, 0], [-2, 1, 0, 3, 2, 3, 2, -3], [0, -2, -2, -5, 0, -2, 7, -7], [4, 5, 1, 0, -1, 3, -3, 10], [3, -4, -9, 9, -1, -2, -3, 2], [3, -5, 3, -3, 0, 0, -5, 5], [-2, 1, 2, -2, 2, 0, 1, 1], [-1, 7, 9, -2, -2, 7, -4, 4], [6, -6, -10, 5, -4, 3, -9, 3], [6, 0, -6, 12, -6, -1, -1, 0], [-1, -4, 5, -5, 2, -2, 6, -5], [1, 5, 0, 0, -1, 5, 0, -5], [-14, 6, 14, -10, 8, -2, 2, 2], [3, -7, -2, 4, 2, 0, -1, -7], [1, 0, 1, -6, -4, 3, 0, 3], [-15, 4, 8, -8, 1, 2, 3, -2], [6, -3, -9, 0, -1, 1, -1, -6], [6, -8, -2, 0, 0, -12, -12, 0], [0, 12, 4, 8, -4, 6, 6, 0], [-1, 6, 7, -4, 6, 3, -4, 5], [-19, 8, 22, -11, 3, 1, 7, 1], [0, -1, -1, 0, -1, -5, 5, -2], [-4, -5, -10, -4, -5, -5, -5, -5], [-2, 1, 5, -5, -1, -2, 2, -3], [10, -2, -9, 11, -8, 0, -10, -2], [-6, -8, -1, 13, -7, -8, 1, -1], [5, 4, 0, 1, -5, 3, -1, -3], [6, -4, -20, 16, -12, -2, -8, 2], [0, 4, 8, -4, 2, -5, -5, 0], [3, -6, 0, 4, -3, -8, -2, 8], [8, -7, -14, 7, -6, 1, -8, 1], [2, -4, -9, -5, -5, 0, 5, -4], [0, -6, -4, -4, -6, 0, 12, -6], [11, 3, -8, 0, -19, 4, -4, 6], [-2, 10, 0, -10, 10, 10, 2, -2], [2, -1, 12, -6, 14, -2, 1, -2], [-2, -2, 0, -8, 2, 4, 6, -4], [-6, 4, -3, 13, 7, 4, -11, 11], [-9, 0, 8, -7, -3, -5, 0, 7], [4, -2, -7, 7, -3, -6, 1, -4], [-2, 0, 3, -6, 1, 5, 0, 4], [14, 2, -21, 19, -23, -8, -3, -3], [24, -5, -30, 15, -6, -3, -2, -3], [-5, 12, 11, -4, 2, 13, 13, 0], [-3, 4, -17, 9, -10, 2, 2, 1], [-2, -11, 0, 0, -2, -10, -1, -10], [8, -1, -9, 0, 0, 0, 0, -2], [-1, 0, -2, -7, -7, -1, 0, -5], [-19, 16, 19, -19, 9, 18, 7, 2], [10, -3, -10, 10, -7, -3, -3, -3], [-16, -5, 0, -10, 16, -4, 1, 4], [2, 2, 15, -9, 12, 1, -6, 4], [8, 1, 0, 18, -8, -2, -3, 2], [7, 10, -8, 4, -1, 4, 6, 4], [5, -1, -6, 0, -1, 10, -10, -2], [-6, 5, 5, -7, 9, 0, -2, 5], [1, 3, 14, -14, 7, 14, -4, 11], [4, 8, 4, 12, -16, 0, 0, 8], [-4, 2, 6, 0, -6, -9, 9, 4], [5, -15, -1, 18, -14, -15, -8, 8], [-10, 7, -10, 5, -20, -5, 12, -5], [-5, 2, 28, -11, 7, 1, 8, -3], [9, -1, 7, -2, -8, -1, -6, 6], [3, -11, -7, -6, 4, 7, -2, -2], [19, -10, 2, -2, -5, -10, -5, 0], [-16, 0, 6, 0, 12, -4, 0, -4], [-21, -4, 20, -25, 24, 4, 0, 0], [-10, 10, 0, 1, -6, 5, 10, 0], [-19, -4, 1, -24, 12, 1, 1, 0], [-1, 16, 7, 8, -9, -2, 7, 7], [-19, 0, 11, -1, 26, 8, 0, 3], [-10, -2, 11, -9, 7, 0, 15, -2], [14, -2, 5, -15, -7, -2, 3, -3], [-11, 10, 0, -10, 11, 11, 1, -11], [-5, 0, 23, -5, 0, 0, 10, -5], [-6, 2, -8, 4, -2, -4, -4, 0], [9, 5, 3, -20, 2, 5, 6, -6], [9, -8, -5, 13, -2, 0, -8, -8], [2, -19, -7, 7, -11, -22, -8, -3], [-10, 3, -20, 20, -3, -6, 6, -9], [7, -10, 0, 14, -3, 0, -1, -10], [-15, -2, -8, -1, 6, -2, 18, -18], [7, 3, -2, 1, 5, 9, -6, 9], [6, -3, 0, 0, -6, 2, 5, -2], [-14, -7, 0, -14, 14, -6, 1, 6], [32, -18, -31, 25, -13, -4, -11, -11], [8, -2, -2, 14, 2, 0, -16, -2], [-6, 0, 4, -10, 6, 10, 0, 2], [-26, 0, 10, -17, 9, 0, 0, -6], [10, 10, 12, 12, 2, 6, 8, 8], [-7, 0, 7, 0, 14, 5, -5, 0], [26, -13, 3, 10, -5, -2, -2, 0], [-1, -7, -6, -2, 1, -2, -2, 0], [-6, 11, 0, 0, -6, 8, 3, 8], [7, 7, 4, 5, -3, 11, 9, 9], [3, -13, 2, -2, -2, -4, -11, 9], [-18, 6, 8, -17, 2, 4, 5, 5], [2, 3, 0, 0, -2, -10, -13, 10], [-21, 14, 21, -28, 14, 5, 5, 0], [-4, 12, -2, 1, -6, -3, 15, -3], [7, -10, -17, 0, 0, 3, -3, -20], [8, -8, -22, -6, -20, 0, 12, -8], [16, -15, -13, 13, -5, -10, -10, 5], [-3, 14, -5, -11, -5, 0, -3, 14], [-5, -4, 1, 0, -15, -12, 12, -8], [20, -6, 0, 16, -20, -2, 4, 2], [31, -16, -23, 38, -19, -8, -8, 0], [4, 6, -16, -8, 22, 6, -2, 2], [22, 0, -16, 15, -8, -1, 0, -10], [-17, 18, 22, -3, 4, -10, 4, 4], [34, -3, -37, 0, -15, 1, -1, -6], [14, -26, -13, 2, -1, -13, -2, -12], [-8, 0, 16, -24, 12, 4, 4, 0], [23, -8, -5, 20, -10, -9, -9, 0], [-3, 4, 18, 4, -5, 2, 2, 1], [11, -2, -13, 0, 24, 7, -7, -4], [23, 0, -13, 12, -12, -1, 0, -3], [9, -1, 9, 27, -14, 0, -3, -1], [23, -21, -29, 16, -8, -7, -14, -14], [-9, 13, 4, -17, 9, 13, 13, -13], [-12, 16, 18, -14, 7, 19, 19, 0], [-20, -4, 0, -16, 20, -2, 2, 2], [6, -12, 12, -6, 18, -14, 2, -14], [-6, -8, 10, 14, -18, -8, 0, 0], [12, -23, -35, -11, 1, 0, -2, -23], [8, -7, -15, 0, 5, -1, 1, -14], [-14, 0, 0, -29, 14, 11, 11, -11], [6, -12, -4, 13, -13, -6, 0, -6], [-8, -9, -15, 9, 6, -9, 8, -8], [2, 9, 12, 8, 3, -3, 3, 3], [14, 0, -14, 14, -18, -18, 0, -14], [-2, 0, 10, -7, -13, -8, 0, 18], [2, 14, 12, -6, 14, 8, 6, 8], [-15, 30, 14, 3, -3, 15, 0, 15], [-6, -21, -19, -8, 4, 0, 0, 0], [0, -4, -18, 9, -18, 3, -7, 3], [-35, 15, 19, -30, 4, 5, 10, 10], [-33, 0, 23, -7, 34, 8, 0, 13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_125_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_125_2_e_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_125_2_e_b(:prec:=8)
    chi := MakeCharacter_125_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_125_2_e_b();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_125_2_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_125_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 10, 26, -10, 1, -10, 11, -5, 1]>],Snew);
    return Vf;
end function;